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Time-fractional nonlinear partial differential equations (TFNPDEs) with proportional delay are commonly used for modeling real-world phenomena like earthquake, volcanic eruption, and brain tumor dynamics. These problems are quite challenging, and the transcendental nature of the delay makes them even more difficult. Hence, the development of efficient numerical methods is open for research. In this paper, we use the concepts of Laplace-like transform and variational theory to develop a new numerical method for solving TFNPDEs with proportional delay. The stability and convergence of the method are analyzed in the Banach sense. The efficiency of the proposed method is demonstrated by solving some test problems. The numerical results show that the proposed method performs much better than some recently developed methods and enables us to obtain more accurate solutions.

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Nonnegative matrix factorization (NMF) is an unsupervised learning method for decomposing high-dimensional nonnegative data matrices and extracting basic and intrinsic features. Since image data are described and stored as nonnegative matrices, the mining and analysis process usually involves the use of various NMF strategies. NMF methods have well-known applications in face recognition, image reconstruction, handwritten digit recognition, image denoising and feature extraction. Recently, several projective NMF (P-NMF) methods based on positively constrained projections have been proposed and were found to perform better than the standard NMF approach in some aspects. However, some drawbacks still affect the existing NMF and P-NMF algorithms; these include dense factors, slow convergence, learning poor local features, and low reconstruction accuracy. The aim of this paper is to design algorithms that address the aforementioned issues. In particular, we propose two embedded P-NMF algorithms: the first method combines the alternating least squares (ALS) algorithm with the P-NMF update rules of the Frobenius norm and the second one embeds ALS with the P-NMF update rule of the Kullback–Leibler divergence. To assess the performances of the proposed methods, we conducted various experiments on four well-known data sets of faces. The experimental results reveal that the proposed algorithms outperform other related methods by providing very sparse factors and extracting better localized features. In addition, the empirical studies show that the new methods provide highly orthogonal factors that possess small entropy values

Nonnegative matrix factorization is a linear dimensionality reduction technique used for decomposing high-dimensional nonnegative data matrices for extracting basic and latent features. This technique plays fundamental roles in music analysis, signal processing, sound separation, and spectral data analysis. Given a time-varying objective function or a nonnegative time-dependent data matrix Y(t), the nonnegative factors of Y(t) can be obtained by taking the limit points of the trajectories of the corresponding ordinary differential equations. When the data are time dependent, it is natural to devise factorization techniques that capture the time dependency. To achieve this, one needs to solve continuous-time dynamical systems derived from iterative optimization schemes and construct nonnegative matrix factorization algorithms based on the solution curves. This article presents continuous nonnegative matrix factorization methods based on the solution of systems of ordinary differential equations associated with time-dependent data. In particular, we propose two new continuous-time algorithms based on the Kullback–Leibler divergence and the Amari α -divergence